Recent Developments in Lie Algebras, Groups and Representation Theory

http://dx.doi.org/10.1090/pspum/086

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American Mathematical SocietyProvidence, Rhode Island

PURE MATHEMATICSProceedings of Symposia in

Volume 86

Recent Developments in Lie Algebras, Groups and Representation Theory

2009–2011 Southeastern Lie Theory Workshop Series

Combinatorial Lie Theory and Applications October 9–11, 2009, North Carolina State University

Homological Methods in Representation Theory May 22–24, 2010, University of Georgia

Finite and Algebraic Groups June 1–4, 2011, University of Virginia

Kailash C. Misra Daniel K. Nakano Brian J. ParshallEditors

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2010 Mathematics Subject Classification. Primary 17B37, 17B55, 17B56, 17B65, 17B67,20C08, 20C11, 20G05, 20G42, 20G43.

Library of Congress Cataloging-in-Publication Data

Recent developments in Lie algebras, groups and representation theory : Southeastern LieTheory Workshop series 2009–2011 : Combinatorial Lie Theory and Applications, October 9–11,2009, North Carolina State University : Homological Methods in Representation Theory, May22–24, 2010, University of Georgia : Finite and Algebraic Groups, June 1–4, 2011, University ofVirginia / Kailash C. Misra, Daniel K. Nakano, Brian J. Parshall, editors.

pages cm. — (Proceedings of symposia in pure mathematics ; volume 86)Includes bibliographical references.

ISBN 978-0-8218-6917-8 (alk. paper)1. Lie algebras–Congresses. I. Misra, Kailash C. (1954–), editor of compilation. II. Nakano,

Daniel K. (Daniel Ken), (1964–), editor of compilation. III. Parshall, Brian (1945–), editor ofcompilation. IV. Southeastern Lie Theory Workshop on Combinatorial Lie Theory and Appli-cations (2009 : North Carolina State University). V. Southeastern Lie Theory Conference onHomological Methods in Representation Theory (2010 : University of Georgia). VI. Southeast-ern Lie Theory Workshop: Finite and Algebraic Groups (2011 : University of Virginia)

QA252.3.R425 2012512′482—dc23

2012026116

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North Carolina State University, October 2009

University of Georgia, May 2010

University of Virginia, June 2011

Contents

Preface ix

Perverse coherent sheaves on the nilpotent cone in good characteristicPramod N. Achar 1

On the vanishing ranges for the cohomology of finite groups of Lie type IIChristopher P. Bendel, Daniel K. Nakano,

and Cornelius Pillen 25

Tilting modules for the current algebra of a simple Lie algebraMatthew Bennett and Vyjayanthi Chari 75

Endotrivial modulesJon F. Carlson 99

Super duality for general linear Lie superalgebras and applicationsShun-Jen Cheng, Ngau Lam, and Weiqiang Wang 113

Structures and representations of affine q-Schur algebrasJie Du 137

Multiplicative bases for the centres of the group algebra and Iwahori-Heckealgebra of the symmetric group

Andrew Francis and Lenny Jones 159

Moonshine paths and a VOA existence proof of the MonsterRobert L. Griess Jr. 165

Characteristic polynomials and fixed spaces of semisimple elementsRobert Guralnick and Gunter Malle 173

“Frobenius twists” in the representation theory of the symmetric groupDavid J. Hemmer 187

The generalized Kac-Wakimoto conjecture and support varieties for the Liesuperalgebra osp(m|2n)

Jonathan Kujawa 201

An approach towards the Kollar-Peskine problem via the Instanton ModuliSpace

Shrawan Kumar 217

On the representations of disconnected reductive groups over Fq

G. Lusztig 227

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viii CONTENTS

Forced gradings in integral quasi-hereditary algebras with applications toquantum groups

Brian J. Parshall and Leonard L. Scott 247

A semisimple series for q-Weyl and q-Specht modulesBrian J. Parshall and Leonard L. Scott 277

Preface

Lie theory represents a major area of mathematical research. Besides its in-creasing importance within mathematics (to geometry, combinatorics, finite andinfinite groups, etc.), it has important applications outside of mathematics (tophysics, computer science, etc.).

During the twentieth century, the theory of Lie algebras, both finite and infinitedimensional, has been a major area of mathematical research with numerous ap-plications. In particular, during the late 1970s and early 1980s, the representationtheory of Kac-Moody Lie algebras (analogs of finite dimensional semisimple Lie al-gebras) generated intense interest. In part, the subject was driven by its interestingconnections with such topics as combinatorics, group theory, number theory, partialdifferential equations, topology and with areas of physics such as conformal fieldtheory, statistical mechanics, and integrable systems. The representation theory ofan important class of infinite dimensional Lie algebras known as affine Lie algebrasled to the discovery of Vertex Operator Algebras (VOAs) in the 1980s. VOAs areprecise algebraic counterparts to “chiral algebras” in two-dimensional conformalfield theory as formalized by Belavin, Polyakov, and Zamolodchikov. These alge-bras and their representations play important roles in a number of areas, includingthe representation theory of the Fischer-Griess Monster finite simple group and theconnection with the phenomena of “monstrous moonshine,” the representation the-ory of the Virasoro algebra and affine Lie algebras, and two-dimensional conformalfield theory.

In 1985, the interaction of affine Lie algebras with integrable systems led Drin-feld and Jimbo to introduce a new class of algebraic objects known as quantizeduniversal enveloping algebras (also called quantum groups) associated with sym-metrizable Kac-Moody Lie algebras. These are q-deformations of the universal en-veloping algebras of the corresponding Kac-Moody Lie algebras, and, like universalenveloping algebras, they carry an important Hopf algebra structure. The abstracttheory of integrable representations of quantum groups, developed by Lusztig, il-lustrates the similarity between quantum groups and Kac-Moody Lie algebras. Thequantum groups associated with finite dimensional simple Lie algebras also havestrong connections with the representations of affine Lie algebras. The theory ofcanonical bases for quantum groups has provided deep insights into the represen-tation theory of quantum groups. More recently, the theory of geometric crystalsintroduced by Berenstein and Kazhdan has opened new doors in representationtheory. In particular, canonical bases at q = 0 (crystal bases) provide a beautifulcombinatorial tool for studying the representations of quantum groups. The quan-tized universal enveloping algebra associated with an affine Lie algebra is calleda quantum affine algebra. Quantum affine algebras quickly became an interesting

ix

x PREFACE

and important topic of research, the representation theory of which parallels thatof the corresponding affine Lie algebras. But the theory is much deeper and richerthan its classical counterpart, providing a clearer picture of connections with theother areas mentioned above.

After the classification of the finite simple groups (now complete), a full un-derstanding of the representation theory of finite simple groups over fields k ofarbitrary characteristic provides a major problem for the 21st century. The spo-radic Fischer-Griess monster (mentioned above) gives one important example of afinite simple group closely related to Lie theory. Apart from the alternating groupsand the 26 sporadic simple groups, the finite simple groups come in infinite familiesclosely related to the finite groups of rational points G(q) of simple algebraic groupsG over algebraically closed fields k of positive characteristic p > 0. (The finite Reeand Suzuki groups are variations on this theme.) The representation theory of thesefinite groups of Lie type thus form a key area of investigation. One can consider afield F , algebraically closed for simplicity, having characteristic , and investigatethe category of FG(q)-representations. There are three cases to consider.

First, in case = 0, take F = C, the complex numbers. This theory is theso-called ordinary representation theory of G(q). As a result of work of Deligne,Lusztig, and many other mathematicians over the past 35 years, the ordinary theoryis quite well understood in comparison to the cases in which > 0.

Second, if = p (the equal characteristic case), take F = k. By work of Stein-berg, the irreducible kG(q)-modules all lift to irreducible rational representationsof the algebraic group G. This fact has provided strong motivation for the study ofthe modular representation theory of the semisimple algebraic groups G over thepast 30 years. For example, a famous conjecture due to Lusztig posits the charac-ters of the irreducible representations when the characteristic p is large (bigger thanthe Coxeter number). For each type, this conjecture has been proved for p “largeenough” by Andersen-Jantzen-Soergel. The proof follows a path from characteristicp to quantum groups at a root of unity to affine Lie algebras and perverse sheaves.Thus, it ultimately involves the infinite dimensional Lie theory discussed above.Although this approach fails to provide effective bounds on the size of the prime p,a new avenue via a related combinatorial category has been recently investigatedby Fiebig. As a result of Fiebig’s work, very large effective bounds for Lusztig’sconjecture are now known. In addition, the determination of the characters forsmall p (i.e., less than the Coxeter number) remains largely uninvestigated.

Third, when 0 < = p (the cross-characteristic case), much less is knownin general. When G is a general linear group GLn(k), the determination of thedecomposition numbers for the finite groups GLn(q) can be determined in termsof decomposition numbers for q-Schur algebras and then for quantum groups overfields of positive characteristic. This is the so-called Dipper-James theory. There areclose connections with the representation theory of Hecke algebras and symmetricgroups. In other types, much less is known; for example, the classification of theirreducible representations is incomplete. A major problem for these other typeswould be to replace the quantum groups used forGLn(q) by some suitable structure.

The modular representation theory has provided a crucial interface with thetheory of finite dimensional algebras (especially, the theory of quasi-hereditary al-gebras introduced by Cline, Parshall and Scott). It seems likely that this direction

PREFACE xi

will continue to prove fruitful. Another significant feature of the modular repre-sentation theory of the finite groups of Lie type and the associated algebraic andquantum groups is the existence of a rich accompanying homological theory. Homo-logical problems emerge immediately because of the failure of complete reducibility.In the equal characteristic case, the homological theory has been extensively de-veloped, for the finite groups of Lie type, quantum enveloping algebras at roots ofunity, restricted Lie algebras and infinitesimal group schemes, as well as other set-tings. Geometric ideas enter via the theory of support varieties, which associate toeach finite-dimensional module for a restricted Lie algebra (or finite group scheme)an algebraic variety. In the cross-characteristic case, much less in known about thecohomology. In the equal characteristic case, there is a considerable body of workinvolving the homological algebra of the infinitesimal groups, and relations betweenthe cohomology of G, its infinitesimal subgroups, and its finite subgroups.

Finally, we mention that the modular representation theory of general finitegroups itself has a strong Lie-theoretic flavor. In part, this is due to the famousAlperin conjecture, suggesting that the irreducible modular representations of gen-eral finite group should be classified in a “weight theoretic” way, much like irre-ducible modules for a complex semisimple Lie algebra are classified by their highestweights. Another notable conjecture, the Broue conjecture, has been recently ver-ified for symmetric groups by Chuang and Rouquier using a the new method of“categorification”.

In 2009, the three editors established a network of Lie theorists in the south-eastern region of the U.S. and proposed an annual regional workshop series of 3 to4 days in Lie theory. The aim of these workshops was to bring together senior andjunior researchers as well as graduate students to build and foster cohesive researchgroups in the region. With support from the National Science Foundation and theaffiliated universities in the region, three successful workshops were held at NorthCarolina State University, the University of Georgia and the University of Virginiain 2009, 2010 and 2011 respectively. Each of these workshops was attended by over70 participants. The workshops included expository talks by senior researchers andafternoon AIM style discussion sessions with a goal to educate graduate studentsand junior researchers in the early part of their study for research in different as-pects of Lie theory. In the third workshop at the University of Virginia, ProfessorLeonard Scott was honored on the eve of his retirement for his lifetime contributionsto many of the aforementioned topics.

The plenary speakers in the three workshops were invited to contribute to thisproceedings. Most of the articles presented in this book are self-contained, andseveral survey articles, by Jon Carlson, Jie Du, Bob Griess, and David Hemmer areaccessible to a wide audience of readers.

The editors take this opportunity to acknowledge the conference participants,the contributors, and the editorial offices of the American Mathematical Societyfor making this volume possible.

Kailash C. MisraDaniel K. NakanoBrian J. Parshall

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Selected Published Titles in This Series

86 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, RecentDevelopments in Lie Algebras, Groups and Representation Theory, 2012

85 Jonathan Block, Jacques Distler, Ron Donagi, and Eric Sharpe, Editors,String-Math 2011, 2012

84 Alex H. Barnett, Carolyn S. Gordon, Peter A. Perry, and Alejandro Uribe,Editors, Spectral Geometry, 2012

83 Hisham Sati and Urs Schreiber, Editors, Mathematical Foundations of QuantumField Theory and Perturbative String Theory, 2011

82 Michael Usher, Editor, Low-dimensional and Symplectic Topology, 2011

81 Robert S. Doran, Greg Friedman, and Jonathan Rosenberg, Editors,Superstrings, Geometry, Topology, and C∗-algebras, 2010

80 D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, and M.Thaddeus, Editors, Algebraic Geometry, 2009

79 Dorina Mitrea and Marius Mitrea, Editors, Perspectives in Partial DifferentialEquations, Harmonic Analysis and Applications, 2008

78 Ron Y. Donagi and Katrin Wendland, Editors, From Hodge Theory to Integrabilityand TQFT, 2008

77 Pavel Exner, Jonathan P. Keating, Peter Kuchment, Toshikazu Sunada,and Alexander Teplyaev, Editors, Analysis on Graphs and Its Applications, 2008

76 Fritz Gesztesy, Percy Deift, Cherie Galvez, Peter Perry, and Wilhelm Schlag,Editors, Spectral Theory and Mathematical Physics: A Festschrift in Honor of BarrySimon’s 60th Birthday, 2007

75 Solomon Friedberg, Daniel Bump, Dorian Goldfeld, and Jeffrey Hoffstein,Editors, Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, 2006

74 Benson Farb, Editor, Problems on Mapping Class Groups and Related Topics, 2006

73 Mikhail Lyubich and Leon Takhtajan, Editors, Graphs and Patterns in Mathematicsand Theoretical Physics, 2005

72 Michel L. Lapidus and Machiel van Frankenhuijsen, Editors, Fractal Geometry andApplications: A Jubilee of Benoıt Mandelbrot, 2004

71 Gordana Matic and Clint McCrory, Editors, Topology and Geometry of Manifolds,2003

70 Michael D. Fried and Yasutaka Ihara, Editors, Arithmetic Fundamental Groups andNoncommutative Algebra, 2002

69 Anatole Katok, Rafael de la Llave, Yakov Pesin, and Howard Weiss, Editors,Smooth Ergodic Theory and Its Applications, 2001

68 Robert S. Doran and V. S. Varadarajan, Editors, The Mathematical Legacy ofHarish-Chandra, 2000

67 Wayne Raskind and Charles Weibel, Editors, Algebraic K-Theory, 1999

66 Robert S. Doran, Ze-Li Dou, and George T. Gilbert, Editors, Automorphic Forms,

Automorphic Representations, and Arithmetic, 1999

65 M. Giaquinta, J. Shatah, and S. R. S. Varadhan, Editors, Differential Equations:La Pietra 1996, 1999

64 G. Ferreyra, R. Gardner, H. Hermes, and H. Sussmann, Editors, DifferentialGeometry and Control, 1999

63 Alejandro Adem, Jon Carlson, Stewart Priddy, and Peter Webb, Editors, GroupRepresentations: Cohomology, Group Actions and Topology, 1998

62 Janos Kollar, Robert Lazarsfeld, and David R. Morrison, Editors, AlgebraicGeometry Santa Cruz 1995, 1997

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/pspumseries/.